Investigation of Matrices Q^(n._L ), M^(n._L ) and R^(n._L ) and Some Related Identities

In this study, it is aimed to use the Lorenz matrix multiplication to find the n^th powers of some special matrices and to reach the quadratic equations and characteristic roots of the matrices obtained in this way. In addition, it is aimed to contribute literature to the studies in the field by reaching some identities.

Keywords:

Characteristic Roots, Fibonacci and Lucas Numbers, Lorentz Matrix Multiplication Identity, Quadratic Equations,

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[1] B.U. Alfred, “An Introduction to Fibonacci Discovery”,The Fibonacci Association, California, 1965.
[2] M. Bicknell, V.E. Hoggatt, “A Primer for the Fibonacci Numbers”, The Fibonacci Quarterly, 1973.
[3] I.D. Ruggles,“Some Fibonacci results using Fibonacci-type sequences”,The Fibonacci Quarterly, vol.1, pp.75-80, 1963.
[4] D. Tasci, E. Kilic,“On the order-k generalized Lucas numbers”,Applied Mathematics and Computation, vol.155, pp.637-641,2004.
[5] E. Kilic, D. Tasci, “On the generalized order-k Fibonacci and Lucas numbers”, Rocky Mountain Journal of Mathematics, vol.36, pp.1915-1926, 2006.
[6] C. Kızılateş, N. Tuglu, “A New generalization of convolved (p,q)-Fibonacci and (p,q)-Lucas polinomials”, Journal of Mathematical and Computational Science, vol.7, pp.995-1005, 2017.
[7] F. Qi, C. Kızılateş, W.S. Du, “A closed formula for the Horadam polynomials in terms of a tridiagonal determinant”, Symettery, vol. 11, pp. 782, 2019.
[8] C. Kızılateş, “New families of Horadam numbers associated with finite operators and their applications”, Mathematical Methods in the Applied Sciences, vol. 44, pp. 14371-14381, 2021.
[9] C. Kızılateş, W.S. Du, F. Qi “Several determinantal expressions of generalized tribonacci polynomials and sequences, Tamkang Journal of Mathematics, vol. 53, pp. 277-291, 2022.
[10] H.W. Gould, “A history of the Fibonacci Q-matrix and a higher-dimensional problem”, Fibonacci Quarterly, vol.9, pp.7-250, 1981.
[11] S. L. Basin, V. E. Hoggatt, “A primer on the Fibonacci sequence”,Part II, Fibonacci Quarterly, vol.2, pp.61-68, 1963.
[12] C.H. King, “Some properties of the Fibonacci numbers”, Master Thesis, San Jose State College, 1960.
[13] J.L. Brenner,“Lucas’ matrix”,The American Mathematical Monthly, vol.58, pp.220-221, 1951.
[14] J.S. Frame, “Continued fractions and matrices”, The American Mathematical Monthly,vol.56,pp.98-103,1949.
[15] H. Schwerdtfeger, “Geometry of complex numbers, mathematical expositions”, University of Totonto Press,1962.
[16] E.Jacobsthal, “Fibonaccische polynome und kreistheilungsgleichungen”,Sitzungsberischte der Berliner Math. Gesellschaft, vol.17 pp. 43-47, 1919-20.
[17] G.K. White, “On generators and defining relations for the unimodular group M2”,The American Mathematical Monthly, vol.71, pp.743-748, 1964.
[18] M.Bicknell, “Fibonacci fantasy:The square root of the Q matrix”,Fibonacci Quarterly, vol.3,pp.67-71, 1965.
[19] T.Koshy, “Fibonacci and Lucas numbers with applications”, Wiley-Interscience Publication, New York, 2001.
[20] H. Gündoğan, O. Keçilioğlu, “Lorentzian matrix multiplication and the motions on Lorentzian plane”,Glasnık Matematic ki, vol.41, pp.329-334, 2006.
[21] J.G. Ratcliffe, “Foundations of Hyperbolic manifolds”,Springer-Verlag, New York, 1994.
[22] O. Keçilioğlu, H. Gündoğan, “Pseudo matrix multiplication”,Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics, vol.66, pp.37-43, 2017.
[23] J.H. Halton, “On a general Fibonacci identity”,The Fibonacci Quarterly, vol.3, pp.31-43, 1965.